Modified Radon Transform for Texture AnalysisVikram Venkatraghavan1, Smruti Rekha1, Jyotirmoy Chatterjee1, Ajoy K. Ray2,3

1School of Medical Science & Technology, I.I.T. Kharagpur, India.2Department of Electronics and Electrical Communication Engineering, I.I.T. Kharagpur, India.

3Bengal Engineering and Science University, Shibpur.

Abstract—The invertibility of Radon Transform is a necessaryproperty in 3-D reconstruction. However it has limited utility whenRadon Transform is used as a texture descriptor. In this paper,we propose a modified non-invertible discrete Radon transform, soas to increase its texture-describing capability. The modification isachieved by incorporating the effect of uniformity of a pixel underconsideration with respect to its neighborhood, on the line integralof the Radon transform. We demonstrate the advantage of sucha modification by showing that the modified Radon transform,when used as a descriptor for content-based image retrieval, showsbetter retrieval performance than the standard Radon transformor its generalization the Trace transform.

I. INTRODUCTION

Texture is a ubiquitous visual experience. It can describea wide variety of surface characteristics. In many machinevision as well as medical imaging applications, study of texturebecomes important as the simplifying assumptions about uni-formity of local image regions often cannot be made. Texture,unlike image intensity, is not defined for a single point inisolation. Although it is widely accepted that texture is a groupphenomenon, a unanimously accepted definition of texture isnot available. Thus, many approaches to texture analysis havebeen proposed in the last three decades. These approaches todeal with texture can be classified into three main groups:Statistical, Structural and Model-Based. A comprehensive re-view of these techniques can be found in [1]. Recently, someresearchers have considered using the Radon transform [2]for texture analysis. The Radon transform is a special case ofprojection-based image analysis. The 2D Radon transform isthe integral transform that computes the integral of a functionalong straight lines. A 2D function can be fully reconstructedfrom the knowledge of its integrals along straight lines definedin its domain. While this knowledge of the existence of theRadon transform′s inverse is useful in the field of 3D recon-struction, it has limited utility in the field of texture analysis.Trace Transform [3], the generalized Radon transform, usesnon-invertible transforms to analyze textures. The Trace trans-form defines various ′functionals′, which operate along straightlines to extract textural features.

From the success of texture descriptors like LawsMask [4],Local Binary Pattern [5], Local Derivative Pattern [6]etc., it can be inferred that neighborhood information arounda pixel is quite important in the field of texture analysis. Boththe Radon and Trace transforms only use gray value of pixelsin the line under consideration while computing the integraland ′functional′ respectively. Our proposed modification to theRadon transform, unlike the traditional Radon transform, not

only considers a line in isolation but also the effect of itsneighboring pixels on it. In doing so, the invertiblity of thetransform is lost. However, the advantage gained for textureanalysis purposes justifies the trade-off. The rest of this paperis organized as follows: section 2 introduces the Radon trans-form and proposes a modified Radon Transform, better suitedfor texture analysis; section 3 demonstrates the experimentalresults and section 4 concludes the paper.

II. THE CONCEPT OF MODIFIED RADON TRANSFORM

A. Radon Transform

Radon transform on a 2-D image J for a given set of anglesmay be obtained by computing the projection of the imagealong the same set of angles. The resulting projection is thesum of the intensities of the pixels in each direction, i.e., a lineintegral. Mathematically, a line (t) can be represented in normalform as:

ρ = x cosβ + y sinβ (1)

where, ρ is the perpendicular distance of the line from theorigin and β is the angle made by the normal of the line withthe x axis. Considering this definition of a line, the Radontransform in 2-D can be defined as in equation 2. An illustrationof the same is shown in figure 1.

R(ρ, β) =

∫ ∞−∞

∫ ∞−∞{J(x, y)δ(ρ− x cosβ − y sinβ)}dxdy

(2)

Fig. 1. An illustration of Radon transform

Once an image is transformed to the radon domain, a de-scriptor is generally used to extract features from that domain,which will facilitate analysis of shape, texture etc. [7], [8], [9]Choice of the descriptor is based on the type of analysis that

needs to be done. For our work on texture analysis, an SVDbased descriptor described in [7] is used.

B. Directional Fuzzy Uniformity Index

1) Concept of Uniformity in Texture Analysis: For the com-putation of uniform LBP (ULBP), Ojala et al. [10] classify eachpixel as uniform or non-uniform. The algorithm used for thispurpose is discussed here:

A circular neighborhood is considered around a pixel. Ppoints are chosen on the circle such that they are all equidistantfrom the center pixel. The gray values at points on the circularneighborhood that do not coincide exactly with pixel locationsare estimated by interpolation. These points are then convertedinto a circular bit-stream of 0s and 1s according to whetherthe gray value of the point is less than or greater than the grayvalue of the center pixel. If the number of bit-transitions in thecircular bit-stream is less than or equal to 2, the center pixel islabeled as uniform. A lookup table is generally used to computethe bit-transitions to reduce computational complexity.

Fig. 2. Illustration of the algorithm used for classification of pixel as uniformor non-uniform

The disadvantages of this method are two-fold:• Generalization to higher dimensions: Classification of a

pixel into the above two classes was then extended in [11]to 3D by computing the number of different islands of1s within the neighbourhood using connected componentlabeling (CCL). However, considering the increase innumber of neighbourhood pixels with dimensions, neitherthe creation the of lookup table nor the CCL itself at eachvoxel, is practical.

• Use of crisp threshold for binarizing: The use of crispthresholds for binarization sometimes leads to misclassi-fications. This is illustrated in figure 3. In Fig. 3(a), thechanges in the region around the center pixel are not muchand yet it is classified as a non-uniform pixel, while that infig. 3(b) is huge, and yet it is classified as a uniform region.

2) Concept of Directional Fuzzy Uniformity Index: To over-come the above mentioned disadvantages, a fuzzy classificationapproach for delineating a pixel based on its uniformity, isintroduced here. This measure, which computes the degree ofuniformity of a pixel, is termed as directional fuzzy uniformityindex (DFUI). In this approach, every pixel is classified as bothuniform as well as non-uniform, but to varying degrees.

Consider a P×Q neighbourhood around a pixel J(x,y) ofthe gray scale image. Consider a line (L) passing through thecentral pixel (x,y) and making an angle θ with the y-axis.Let N be the number of points that we consider within the

Fig. 3. Illustration of cases where the traditional algorithm fails

P×Q neighbourhood which falls on L. Let J1, J2, . . . ,JN be theintensities of the points under consideration. The gray valuesof points which do not fall exactly in the centre of pixels areestimated by interpolation. The degree of uniformity of thecenter pixel in the direction θ is given by:

DFUI(x, y; θ) =f(|Ji − J(x, y)|)

N(3)

where, f(|z|) is a monotonically decreasing fuzzy membershipfunction, and f(0) = 1.

C. Proposed Modification in Radon Transform

The traditional radon transform computes the line integralby giving equal weightage to the all the pixels on the line. Weargue that for texture analysis, the pixels which are presentin a ′uniform′ region should have much more effect on theline integral than the pixels of the same gray value in a ′non-uniform′ region, since the former is representative of the localgray values in that region while the later is not. Accordingly,we formulate the modified radon transform on an image J(x,y)as shown in equation 4.

R̂(ρ, β) =

∫ ∞−∞

∫ ∞−∞{J(x, y)(1 +DFUI(x, y))

δ(ρ− x cosβ − y sinβ)}dxdy (4)

It can be inferred from equation 4 that the weightage given toa pixel during the line integral is proportional to the degreeof uniformity of the pixel in the defined neighborhood. Now,suppose G(x, y) = J(x, y)× (1+DFUI(x, y)). It is possibleto get G(x,y) from modified Radon transform R̂ but J(x,y)cannot be obtained from G(x,y), since f : J →G is a many-to-many function. Hence on the whole, R̂ is a non-invertibletransform.

III. EXPERIMENTS

A. Effect of Additive Noise on DFUI

Brodatz’s texture database [12] was used for studying theeffect of noise on DFUI . Additive White Gaussian Noisewas progressively added to study the variations in DFUIhistograms of various textures. Figure 4 shows the changein distribution of the histogram of DFUI when noise wasadded in the texture. The membership function used for the

TABLE IVARIATION OF PERCENTAGE OF PIXELS ABOVE 0.5 DFUIG40(75◦) WITH

NOISE

Noise Added Percentage of pixelsNoiseless 84.7%

25dB 70.2%20dB 51.5%15dB 34.6%

experiment was a zero mean Gaussian membership functionwith a variance of 30. DFUI computed using this member-ship for theta = 35◦ is denoted as DFUIG30(35

◦). Simi-lar effects were seen in DFUI histograms constructed usingvarious membership functions for various textures. It was thusinferred from the experiment that a noiseless texture has morepercentage of pixels with high DFUI . Gradual addition ofnoise makes the histogram shift gradually towards the left end.Thus the modified Radon transform, in effect, suppresses theeffect of noise on the texture descriptor computation by givingweightage to each pixel in the line integral calculation based onthe degree of uniformity of the pixel.

(a) (b) (c)

Fig. 4. (a) D11 Texture in the Brodatzs texture database. (b) DFUIG30(35◦)histogram for D11 texture (c)DFUIG30(35◦) histogram for D11 texture afteradding 15dB of noise.

Percentage of pixels in the entire Brodatz’s texture database,with a DFUIG40(75

◦) value of 0.5 or more was calculatedwhen AWGN was added progressively to the images and hasbeen tabulated in Table I.

B. An Application to Image Retrieval

A database with 94 types of fishes [3] was taken for the caseof image database retrieval. Fig. 5 shows some of the imagestaken from the database. SVD-based features [7] computedusing the Radon and Modified Radon transform were extractedfrom the images in the database and the query image. Thisfeature set was then used as the criterion for matching in imageretrieval. The method developed here was also compared withthe trace transform-based descriptors developed in [3].

1) Experiment:Corrupted Image Query: AWGN was addedto each object in the database and the corrupted image was usedas the query image to check the robustness to noise. The resultsof these experiments are tabulated in Table II. The numbersin the second, third and fourth column in each of the tablesrepresent the number of images that were retrieved correctly asthe first choice.

2) Experiment:Corrupted Database: Additive white Gaus-sian noise was added to each image in the database and thecorrupted images were added to the existing database. The

Fig. 5. Some of the images from the database

TABLE IIROBUSTNESS IN PRESENCE OF ADDITIVE GAUSSIAN NOISE

NoiseAdded(dB)

Modified RadonTransform basedSVD feature

Radon Transformbased SVD fea-ture

Trace TransformTriplefeature(Π1) [3]

30 94 94 9425 94 94 9420 93 85 8815 73 45 49

noises added in the database were: 30dB, 25dB and 20dB.Each Image was also rotated by angles 15◦, 30◦, 45◦&60◦ andadded to the database. Each image in the newly formed databasewas then used as query image and the retrieved images wereanalyzed for evaluation. Fig. 6 shows a sample query along withsome of the retrieved images.

Fig. 6. (a) Sample Query image (b)-(d) Retrieved images with ranks 1, 6 and13 respectively.

Table III shows the average number of relevant images re-trieved in top 15 hits for all 3 methods.

The number of images retrieved with each query was thenmethodically varied to study the performance of each algo-

TABLE IIIOVERALL AVERAGE NUMBER OF RELEVANT HITS IN TOP 15

Method Overall AverageRadon Transform + SVD 11.7542Modified RadonTransform + SVD

12.8431

Trace Transform triplefeature(Π1) [3]

12.1760

rithm. Precision and recall were calculated using equations 5and 6 respectively.

Precision =No.of Correctly Retrieved Images

No.of Retrieved Images(5)

Recall =No.of Correctly Retrieved Images

No.of Relevant Images(6)

It can be interpreted from Fig. 7 that recall is considerablyhigher for the same amount of precision for the proposedmethod when compared with the Radon and Trace transform-based methods. Hence, from the above performed experiments,it can hence be concluded that the modified Radon transform,when used for texture description in content based image re-trieval, shows better retrieval performance than the standardRadon transform or its generalization, the Trace transform.

Fig. 7. 1 - Precision vs Recall graph

IV. CONCLUSION AND FUTURE WORK

We have presented a new method for modifying RadonTransform to make it more suited to texture analysis. Byincorporating uniformity information within the computationof the line integral, we add local neighborhood informationto the global information that Radon Transform extracts. Thecombination of the two is then proved to be more suited fortexture analysis by showing its performance to be better ina content-based image retrieval problem. The current workon modification of radon transform can be extended easily tomodify Trace transform [3] and Ridgelets [13] using similardefinitions. The fuzzy solution to classification of a pixel asuniform or non-uniform, as proposed in this paper solvesvarious problems that the previous definition and its exten-sions had. The concept of uniformity in texture analysis hastraditionally been used for computing Uniform Local BinaryPattern (ULBP) [10]. The proposed solution for classifying apixel as uniform or non-uniform can be easily extended tohigher dimensions, thus making uniformity as well as textureanalysis using ULBP more computationally efficient comparedto the existing method in [11]. The extension of the concept ofuniformity to dynamic textures [14] can also be considered as apossible direction for future work.

REFERENCES

[1] M. Tuceryan and A. K. Jain, “Texture analysis,” in Handbook of patternrecognition & computer vision, C. H. Chen, L. F. Pau, and P. S. P. Wang,Eds. River Edge, NJ, USA: World Scientific Publishing Co., Inc., 1993,pp. 235–276.

[2] K. Jafari-Khouzani and H. Soltanian-Zadeh, “Rotation-invariant mul-tiresolution texture analysis using radon and wavelet transforms,” ImageProcessing, IEEE Transactions on, vol. 14, no. 6, pp. 783 –795, june2005.

[3] A. Kadyrov and M. Petrou, “The trace transform and its applica-tions,” Pattern Analysis and Machine Intelligence, IEEE Transactions on,vol. 23, no. 8, pp. 811 –828, aug 2001.

[4] K. Laws, “Textured image segmentation,” in USC ISG, 1980.[5] T. Ojala, M. Pietikainen, and D. Harwood, “A comparative study of tex-

ture measures with classification based on feature distributions,” PatternRecognition, vol. 29, no. 1, pp. 51–59, jan. 1996.

[6] B. Zhang, Y. Gao, S. Zhao, and J. Liu, “Local derivative pattern versuslocal binary pattern: Face recognition with high-order local pattern de-scriptor,” Image Processing, IEEE Transactions on, vol. 19, no. 2, pp.533 –544, feb. 2010.

[7] O. Al-Shaykh and J. Doherty, “Invariant image analysis based on radontransform and svd,” Circuits and Systems II: Analog and Digital SignalProcessing, IEEE Transactions on, vol. 43, no. 2, pp. 123 –133, feb 1996.

[8] V. Venkatraghavan, M. Agarwal, and A. Roy, “Shape orientation pattern,”Signal Processing Letters, IEEE, vol. 16, no. 8, pp. 711 –714, aug. 2009.

[9] G. Liu, Z. Lin, and Y. Yu, “Radon representation-based feature descriptorfor texture classification,” Image Processing, IEEE Transactions on,vol. 18, no. 5, pp. 921 –928, may 2009.

[10] T. Ojala, M. Pietikainen, and T. Maenpaa, “Multiresolution gray-scale androtation invariant texture classification with local binary patterns,” PatternAnalysis and Machine Intelligence, IEEE Transactions on, vol. 24, no. 7,pp. 971 –987, jul 2002.

[11] L. Paulhac, P. Makris, and J.-Y. Ramel, “Comparison between 2d and3d local binary pattern methods for characterisation of three-dimensionaltextures,” in Proceedings of the 5th international conference on ImageAnalysis and Recognition, ser. ICIAR ’08. Berlin, Heidelberg: Springer-Verlag, 2008, pp. 670–679.

[12] P. Brodatz, in Textures: A photographic album for artists and designers.New York: Dover Publications, 1966.

[13] K. Huang and S. Aviyente, “Rotation invariant texture classification withridgelet transform and fourier transform,” in Image Processing, 2006IEEE International Conference on, oct. 2006, pp. 2141 –2144.

[14] G. Zhao and M. Pietikainen, “Local binary pattern descriptors for dy-namic texture recognition,” in Pattern Recognition, 2006. ICPR 2006.18th International Conference on, vol. 2, 2006, pp. 211 –214.